Numerical approximation of a PDE-constrained Optimization problem that appears in Data-Driven Computational Mechanics

TL;DR

This paper develops a finite element approach for solving a PDE-constrained optimization problem in Data-Driven Computational Mechanics, ensuring stability and preserving the problem's saddle-point structure, with demonstrated numerical effectiveness.

Contribution

It introduces a stable finite element discretization for a PDE-constrained optimization in Data-Driven Mechanics, maintaining the saddle-point structure and allowing equal-order interpolation.

Findings

The method is stable and convergent.

It accurately approximates the continuous problem.

Numerical examples confirm effectiveness.

Abstract

We investigate an optimization problem that arises when working within the paradigm of Data-Driven Computational Mechanics. In the context of the diffusion-reaction problem, such an optimization problem seeks for the continuous primal fields (gradient and flux) that are closest to some predefined discrete fields taken from a material data set. The optimization is performed over primal fields that satisfy the physical conservation law and the geometrical compatibility. We consider a reaction term in the conservation law, which has the effect of coupling all the optimality conditions. We first establish the well-posedness in the continuous setting. Then, we propose stable finite element discretizations that consistently approximate the continuous formulation, preserving its saddle-point structure and allowing for equal-order interpolation of all fields. Finally, we demonstrate the effectiveness of the proposed methods through a set of numerical examples.